On parameters of orders of quartic fields and essential pairs -- extended version

Abstract

Classes of pairs of ternary quadratic forms parametrize quartic rings by a result of Bhargava. We give an algorithm for finding a pair of ternary quadratic forms that parametrize a given order of a quartic field. We examine a new technique, essential pairs, for obtaining parameters of orders of quartic fields from a number field database. Essential pairs for maximal orders of quartic fields are very common and provide a simple means of obtaining an integral basis for the ring of integers.

This is the extended version by including proofs of the algorithm and the lemma supporting the algorithm, and by supplying addition results on essential pairs